Property (T) was introduced in the mid 60’s by D. Kazhdan, as a tool to demonstrate that a large class of lattices are ﬁnitely generated. The discovery of Property (T) was a cornerstone in group theory and the last decade saw its importance in many diﬀerent subjects like ergodic theory, abstract harmonic analysis, operator algebra and some of the very recent topics like C*-tensor categories. In the late 1980’s the subject of operator algebraic quantum groups gained prominence starting with the seminal work of Woronowicz, followed by works of Baaj, Skandalis, Woronowicz, Kustermans, Vaes and others. Quantum groups can be looked upon as noncommutative analogues of locally compact groups and in this sense it was quite natural to explore the possibility of extending the notion of Property (T) to the realm of quantum groups. This was done in the following sequence: Property (T) was ﬁrst studied within the framework of Kac algebras (a precursor to the theory of locally compact quantum groups), then for algebraic quantum groups and discrete quantum groups, and ﬁnally for locally compact quantum groups by Joita, Petrescu, Fima, Soltan, Kyed, Skalski, Viselter, Daws, Brannan and Kerr. Thus far, Property (T) was studied only for quantum groups with trivial scaling automorphism group. Quite recently we found a way of extending all the above studies to quantum groups with non-trivial scaling automorphisms.
This talk will be a summary of the results that we have. We will start from the situation with groups abd then move over gradually to the quantum groups.

We will review the current knowledge about monotone convolutions, monotone infinite divisibility, and monotone increment processes. The talk will close with several conjectures and open problems. Based on joint work with Takahiro Hasebe, Ikkei Hotta, and Sebastian Schleissinger.

Thursday, 09-11-2017 - 10:00, 604

Bi-monotone independence

Malte Gerhold (Ernst-Moritz-Arndt-Universität Greifswald)

A new notion of independence for pairs of noncommutative random variables will be discussed. As for bi-freeness, which comes from a simultaneous "left" and "right" realization of freeness on the free product of Hilbert spaces, bi-monotone independence comes from the simultaneous study of monotone and antimonotone independence on the tensor product of Hilbert spaces. We will present a central limit theorem which admits a combinatorial description using bi-monotone pair partitions.

Distributions on non-symmetric position operators on Weakly Monotone Fock Space.

Maria Elena Griseta (University of Bari Aldo Moro)

In this talk we investigate the distributions for sums of random variables $x_i={a_i}^{-}+{a_i}^{†}+\lambda {a_i}^0$, where ${a_i}^{-}$, ${a_i}^{†}$ and ${a_i}^0$ denote respectively annihilation, creation and conservation operators on the Weakly Monotone Fock Space and $\lambda\in \mathbb{R}$.
We start considering sum of position operators (case $\lambda=0$): after obtaining a recursive formula for the moments
\[
\mu_{m,n} := \omega_{\Omega}\bigg(\bigg(\sum_{k=1}^{m}(A_{k}+A_{k}^{†})\bigg)^{2n}\bigg),
\]
where $\Omega=1\oplus 0\oplus 0 \oplus 0\oplus \ldots$ is the vacuum vector and $\omega_{\Omega}(\cdot)=\left\langle \Omega,\cdot\Omega\right\rangle$ is the vacuum expectation, we calculate the Cauchy Transform of the distribution measure and the density for $m=2$.
Finally we compute the distribution of a single random variable $x_i$ with respect to the vacuum state for the general case $\lambda\neq 0$.

We present Poisson type limit theorems for a noncommutative independence
(the bm-independence), which is naturally associated with positive
symmetric cones in euclidian spaces, including R_+^d, the Lorentz cone in
Minkowski spacetime and positive definite (real symmetric or complex
hermitian) matrices. The geometry of the cones plays significant role in
the study as well as the combinatorics of bm-ordered partitions.

Abstract: I will introduce a notion of Coxeter group associated to a finite family of fields, (e.g. generators of nilpotent Lie groups), present examples and possibly interesting questions to Algebraists and Analysts. I will also introduce natural Dunkl type extensions of fields and Markov semigroups, and present some (crude) bounds of corresponding heat kernel.

Part.II Smoothing and Ergodicity of Dissipative Dynamics for Large Interacting Systems

Abstract: This would be about Properties of Markov Semigroups with Hormarder type generators as well as some generalisation of Dunkl generators.

Thursday, 11-05-2017 - 10:15, 604

Spectral theory of Fourier-Stieltjes algebras

Przemysław Ohrysko (IMPAN, Warszawa)

My talk is devoted to presentation of the most important results from a
joint work with Mateusz Wasilewski concerning spectral properties of
Fourier-Stieltjes algebras. It is an extensive project including research
on elements and Gelfand spaces of the discussed algebras. The most
signicant topics are concentrated around the notion of the spectrum of an
element in a relation to an image of a function on a group which leads to
the notion of the naturality of the spectrum and Wiener-Pitt phenomenon
(for non-commutative groups). A lot of other patologies can be transferred
from the commutative case but some problems do not have the classical
counterparts which is caused by the non-commutativity of group C*
-algebras. The subject matter is very broad and the most of the results
will only be outlined. Nevertheless I will try to present in an accesible
way the dierences between the world of measures on commutative (locally
compact) groups and Fourier-Stieltjes algebras (for now only for discrete
groups).

Thursday, 27-04-2017 - 10:15, 604

Spectral properties of some ensembles of random matrices

Karol Życzkowski (IF UJ & CFT PAN)

We analyze spectral density of ensembles of positive hermitian random matrices related to free convolutions of Marchenko-Pastur distributions. Furthermore, we study asymptotic support of the spectrum and numerical range (field of values) for ensembles of non-hermitian random matrices with independent Gaussian entries.

We investigate the existence of a priori estimates for differential
operators in the L1 norm: for anisotropic homogeneous differential
operators T1 , . . . , Tl , we study the conditions under which the
inequality
l\||T_1f\|_1 < \|T_2f\|_1+\|T_3f\|_1+ ......+\|T_lf\|_1
holds true. Properties of homogeneous rank-one convex functions play the
major role in the subject. We generalize the notions of quasi- and
rank-one convexity to fit the anisotropic situation.
We also discuss a similar problem for martingale transforms and provide
various conjectures.

Thursday, 30-03-2017 - 10:15, 604

Complete metric approximation property of q-Araki-Woods algebras

Mateusz Wasilewski (IMPAN)

We will prove that q-Araki-Woods algebras, which are constructed
as a combination of two deformations of the free group factors, have the
complete metric approximation property. In the proof we transfer the result
from the q-Gaussian algebras (one of the deformations mentioned
previously) using a certain central limit theorem and ultraproduct techniques.

Tuesday, 28-03-2017 - 12:15, WS

Symmetric spaces and K-theory

Wend Werner (Universität Münster)

Roughly a quarter of (Riemannian) symmetric spaces are
hermitian and of non-compact type. Each such manifold
carries an algebraic structure on its tangent bundle
which is similar to (more general than) the algebraic
structure of a C*-algebra.
We exploit this similarity in order to apply K-theoretical
methods to the classification of these manifolds. Whereas
this technique reproduces well-known results in finite
dimensions, it is still viable for infinite dimensional
manifolds and can here be used to e.g. give a
K-theoretical classification of inductive limits of bounded
symmetric domains.

Thursday, 23-03-2017 - 10:15, 604

Non-commutative separate continuity: Ellis joint continuity theorem

Biswarup Das (University of Oulu, Finland)

Let S be a topological space, which is also a semigroup with identity,
such that the multiplication is separately continuous. Such semigroups
are called semitopological semigroups. These type of objects occur
naturally, if onestudies weakly almost periodic compactification of a
topological group.
Now if we assume the following:
(a) The topology of S is locally compact.
(b) Abstract algebraically speaking, S is a group (i.e. every element has
an inverse).
(c) The multiplication is separately continuous as above (no other
assumption. This is the only assumption concerning the interaction of the
topology with the group structure).
Then it follows that S becomes a topological group i.e. :
(a) The multiplication becomes jointly continuous.
(b) The inverse is also continuous.
This extremely beautiful fact was proven by R. Ellis in 1957 and is known
in the literature as Ellis joint continuity theorem.
In this talk, we will prove a non-commutative version of this result. Upon
briefly reviewing the notion of semitopological semigroup, we will
introduce ''compact semitopological quantum semigroup'' which were before
introduced by M. Daws in 2014 as a tool to study almost periodicity of
Hopf von Neumann algebras. Then we will give a necessary and sufficient
condition on these objects, so that they become a compact quantum group.
As a corollary, we will give a new proof of the Ellis joint continuity
theorem as well.
This is the joint work with Colin Mrozinski.

The goal of this talk is to recall a noncommutative Brouwer fixed-point
theorem, and show in detail how it is a special case of the torsion
noncommutative Borsuk-Ulam theorem. In particular, the issue of
non-contractibility of compact quantum groups shall be explored.

The Littlewood-Richardson process is a discrete random
point process which encodes the isotypic decomposition of tensor
products of irreducible rational representations of GLN(C).
Biane-Perelomov-Popov matrices are a family of quantum random
matrices arising as the geometric quantization of random
Hermitian matrices with deterministic eigenvalues and uniformly
random eigenvectors. As first observed by Biane, correlation
functions of certain global observables of the LR process
coincide with correlation functions of linear statistics of sums
of classically independent BPP matrices, thereby enabling a
random matrix approach to matrix approach to the statistical
study of GLN(C) tensor products. In this paper, we prove an
optimal result: classically independent BPP matrices become
freely independent in any semiclassical/large-dimension limit.
This removes all assumptions on the decay rate of the
semiclassical parameter present in previous works, and may be
viewed as a maximally robust geometric quantization of
Voiculescu's theorem on the asymptotic freeness of independent
unitarily invariant random Hermitian matrices. In particular,
our work proves and generalizes a conjecture of Bufetov and
Gorin, and shows that the mean global asymptotics
of GLN(C) tensor products are governed by free probability in
any and all GLN(C) tensor products are governed by free
probability in any and all semiclassical scalings. Our approach
extends to global fluctuations, and thus yields a Law of Large
Numbers for the LR process valid in all semiclassical scalings.

Thursday, 26-01-2017 - 10:15, 604

On the distributions of position operators on Weakly Monotone Fock Space

Maria Elena Griseta (University of Bari Aldo Moro)

In this talk we investigate the distributions for sums of creation
and annihilation operators on Weakly Monotone Fock Space.
After obtaining a recursive formula for the moments $\mu_{m,n} :=
\omega_{\Omega}((\sum_{k=1}^{m}(A_{k}+A_{k}^{\dagger}))^{2n})$, where
$\Omega$ is the vacuum vector and $\omega_{\Omega}$ is the vacuum
expectation, we calculate the Cauchy Transform of the distribution measure.

Tuesday, 24-01-2017 - 12:15, WS

Distance matrices and quadratic embedding of graphs

Nobuaki Obata (Tohoku University)

General criteria for a graph to admit a quadratic embedding
are discussed and, as a quantitative approach, the "QE constant"
is
introduced. Concrete examples are obtained from well-known
graphs with graph operations and the QE constants are determined
for all graphs on $n$ vertices, $n\le5$.

We present Poisson type limit theorems for a noncommutative independence
(the bm-independence), which is naturally associated with positive
symmetric cones in euclidian spaces, including $\R_+^d$ and the Lorentz
cone in Minkowski spacetime. The geometry of the cones plays significant
role in the study as well as the combinatorics of bm-ordered partitions.

Here we deal with the sample covariance (SC) matrices of the form M=YY^T,
where columns y_i, i=1,...,m of the matrix Y are independent random vectors
in R^n. Under the assumption that all entries of Y are independent, the
asymptotic spectral analysis of SC matrices has been actively developed
since the celebrated work of Marcenko and Pastur (1967). Less is known
about large SC matrices with dependent entries in columns of Y. In papers
by Bai and Zhou, Pajor and Pastur, and Yaskov, there were considered
limiting spectral distributions of SC matrices with some general
dependence structures of y_i. The next natural step is to study asymptotic
fluctuations of linear eigenvalue statistics of the form
Tr f(M), where f is a test function. We show that if y_1,..., y_m are
i.i.d. normalized isotropic random vectors satisfying certain moment
conditions, then in the limit when m,n tend to infinity and m/n tends to
c>0, the centered linear eigenvalue statistics converge in distribution to
a Gaussian random variable.

Thursday, 08-12-2016 - 10:15, 604

The amenability from algebraic and analytical perspective

Rachid El Harti (Université Hassan I FST de Settat, Maroko)

In this talk, we investigate the amenability from the
algebraic and analytical point of view and its relationship with the
semisimplicity in the case of operator algebras and cross product Banach
algebras associeted with a class of C*-dynamical systems.

Thursday, 17-11-2016 - 10:15, 604

bm-Central Limit Theorem associated with nonsymmetric cone

Oussi Lahcen (Hassan I University, Maroko)

We formulate and prove the general version of the bm-Central
Limit Theorem (CLT) for bm-independent random variables associated with
nonsymmetric cones, in particular circular cone
$\mathcal{C}_{\theta}^{n}$, such sector $\Omega_{u}^{d}\subset
\mathbb{R}^{d}$ and also the Vinberg's cone $\Pi_{v}$ is studied.

Thursday, 10-11-2016 - 10:15, 604

Direct product of automorphism groups of digraphs

Anna Krystek, Łukasz Wojakowski

The problem of representability of a permutation group A as the full automorphism group of a digraph G = (V, E) was first studied for regular permutation groups by many authors, the solution of the problem for undirected graphs was first completed by Godsil in 1979. For digraphs, L. Babai in 1980 proved that, except for the groups S_2^2 , S_2^3 , S_2^4 , C_3^2 and the eight element quaternion group Q, each regular permutation group is the automorphism group of a digraph. Later on, the direct product of automorphism groups of graphs was studied by Grech. It was shown that, except for an infinite family of groups S_n × S_n , n ≥ 2, and three other groups D_4 × S_2 , D_4 × D_4 , and S_4 × S_2 × S_2, the direct product of automorphism groups of two graphs is, itself, an automorphism group of a graph. We study the direct product of automorphism groups of digraphs. We show that, except for the infinite family of permutation groups S_n × S_n , n ≥ 2 and four other permutation groups D_4 × S_2 , D_4 × D_4 , S_4 × S_2 × S_2 , and C_3 × C_3 , the direct product of automorphism groups of two digraphs is itself the automorphism group of a digraph.

Thursday, 03-11-2016 - 10:15, 604

Noncommutative Borsuk-Ulam-type conjecture revisited

Ludwik Dąbrowski (SISSA Trieste/IMPAN) and Piotr M. Hajac (IMPAN)

Let H be the C*-algebra of a non-trivial compact quantum group acting
freely on a unital C*-algebra A. Baum,
Dąbrowski and Hajac conjectured that there does not exist an
equivariant *-homomorphism from A to the equivariant
noncommutative join C*-algebra A*H. When A is the C*-algebra of
functions on a sphere, and H is the C*-algebra of
functions on Z/2Z acting antipodally on the sphere, then the
conjecture becomes the celebrated Borsuk-Ulam Theorem.
Recently, Chirvasitu and Passer proved the conjecture when H is
commutative. In a simple way, we extend this result to
a far more general setting assuming only that H admits a character
different from the counit. We show how our result implies a
noncommutative Brouwer fixed-point theorem and, in particular, the
non-contractibility of such compact quantum groups. Moreover, assuming
that our compact quantum group is a q-deformation of a compact
connected semisimple Lie group, we prove that there exists a
finite-dimensional representation of the compact quantum group such
that, for any C*-algebra A admitting a character, the finitely
generated projective module associated with A*H via this
representation is not stably free. Based on joint work with Sergey
Neshveyev.

We define spreadability systems as a generalized notion of independence,
extending exchangeability systems, to unify various notions of
cumulants known in noncommutative probability.
To this end we study generalized zeta and M\"obius functions in
the context of the incidence algebra of the semilattice of
ordered set partitions. The combinatorics are intimately related
to the coefficients of the Campbell-Baker-Hausdorff formula and
the latter can be seen as a special case of a particular
spreadability system.

We will talk about a class of von Neumann algebras introduced by Hiai. They
form a combination of two generalisations of free group factors: q-Gaussian
algebras of Bożejko and Speicher, and deformations of free group factors
introduced by Shlyakhtenko. We will prove that they always possess the
Haagerup approximation property and we will discuss what is not yet known
about these algebras.

Thursday, 07-04-2016 - 12:15, 604

New class of idempotent Fourier multipliers on $H^1(\mathbb{T}\times\mathbb{T})$

Maciej Rzeszut (IMPAN)

The classification of idempotent Fourier multipliers on $H^1(\mathbb{T})$ is known since 1987, due to a theorem of Klemes. Our main result is an example of an idempotent Fourier multiplier on $H^1(\mathbb{T}\times\mathbb{T})$ that is not dervied by manipulation of tensor products of one-dimensional case. The main tool is a new $L^1$ equivalent of the Stein martingale inequality which holds for a special filtration of periodic subsets of $\mathbb{T}$ with some restrictions on the functions involved. We also identify the isomorphic type of the range of the associated operator as the independent sum of dyadic $H^1_n$.

An important role in topology is played by absolute retracts and
absolute neighborhood retracts that, roughly speaking, are spaces with good
extension properties. In the noncommutative world, the role of spaces with
good extension properties is played by C*-algebras with good lifting
properties, i.e. projective and semiprojective C*-algebras. Some statements
about absolute retracts and neighborhood retracts might be false in the
noncommutative setting or hard to prove. We are going to discuss some
problems of this sort and their connection with problems in Operator
Theory, namely with a question of Akemann, Olsen and Pedersen on best
approximation by compact operators. This is joint work with Terry Loring.